Generalization of the concept of convergence over any Field $\mathbb F$ 
"Let $(a_n)_{n \in \mathbb N}$ be a sequence of elements of $\mathbb R$ or $\mathbb C$."

All of the sequences that I've studied until this point were sequences of this kind, and I have an intuitive understanding of what convergence means in this context. But that got me wondering: What if instead of considering sequences of Real or Complex numbers we consider sequences of elements of a Field $\mathbb F$?
Is there any natural way to extend the definition of convergence to any field? I know that some fields can be very exotic and strange so, if the concept of convergence doesn't make sense when we are working over a general Field, what properties must a field $\mathbb F$ have in order for convergence to make sense?
For example, does convergence of a sequence over $\mathbb Q$ make sense?
 A: Convergence makes intuitive sense as soon as one has a metric on some set $X$. A metric is just a function $d: X \times X \longrightarrow \mathbb{R}_{\geq 0}$, "measuring the distance" between two elements of $X$. To be considered a metric, $d$ must satify the following 3 ($\forall x,y,z \in X$):

*

*$d(x,y) = 0 \iff x = y$

*$d(x,y) = d(y, x)$

*$d(x,z) \leq d(x,y) + d(y,z)$
Now for a sequence $(x_n)_n \subset X$, one says that the sequence converges if there is $x\in X$ such that $\forall \epsilon > 0$ we find some $n_0 \in \mathbb{N}$ such that $d(x_n,x) < \epsilon \; \forall n \geq n_0$.
Intuitively it says that, given any small radius around the limit point, we always find a "timepoint" where all elements of the sequence $(x_n)$ stay inside this small radius. Note that this notion is completely indepentend of any field. But it makes sense to define convergence on for instance normed vectorspaces over some field, such as $\mathbb{R}, \mathbb{C}$ or $\mathbb{R}^n$.
If we have a norm, "measuring distance from the origin", we can define a metric on the space, by setting $d(x,y) := ||x-y||$.
It is possible to define convergence more generally, using topologies as mentioned before. The idea is basically the same, one just has a different notion on what a "small radius" should be using the concept of open subsets.
